J´ozsef Wildt International Mathematical Competition

Vol. 23, No.1, April 2015, pp 184-192

Print: ISSN 1222-5657, Online: ISSN 2248-1893 http://www.uni-miskolc.hu/∼matsefi/Octogon/

184

J´ ozsef Wildt International Mathematical Competition

The Edition XXVI^th , 2016²²

The solution of the problems W.1 - W.50 must be mailed before 10.

September 2016, to Mih´aly Bencze, str. H˘armanului 6, 505600 S˘acele - N´egyfalu, Jud. Bra¸sov, Romania, E-mail: benczemihaly@yahoo.com;

benczemihaly@gmail.com W1. Show that H =�

q odd, squarefreeq≤Q

�

p|q 2

p−2 ≫(logQ)².

Michael Rassias W2. Let D be afixed squarefree integer withD≥1. Let also

πD(x) =�

p≤x

�−D p

�=1

1.

Prove thatπD(x)∼ ¹₂π(x), when x→+∞

Michael Rassias W3. Let (F_k)_k≥0, F0= 0, F1 = 1, F_k+2=F_k+F_k+1, (∀)k∈N and

A= (aij)1≤i≤m

1≤j≤n , B= (bjk)1≤j≤n

1≤k≤m, C= (crs)_1≤r,s≤m, aij =Fj, bjk =Fj, crs=F_m−r+1² ,(∀)i, k= 1, m, (∀) 1, n. Forp, q positive integers compute (AB)^pC^q.

D.M.B˘atinet¸u-Giurgiu and Neculai Stanciu W4. Let a, b∈(−∞,+∞). Find lim

n→∞

�(n+3)^n+2+a

(n+2)^n+1+b − ⁽ⁿ⁺²⁾_(n+1)^n+1+an+b

� .

D.M.B˘atinet¸u-Giurgiu and Neculai Stanciu

22Received 15.03.2016

2000Mathematics Subject Classification. 11-06.

Key words and phrases. Contest.

W5. Let f :R^∗₊→R^∗₊be a continue function such that lim

x→∞

f(x)

x^m =a∈R^∗₊, wherem∈[1,+∞). Calculate lim

n→∞

�

n+1

�n+1�

k=1

f(k)− ⁿ

��n

k=1

f(k)

� .

D.M.B˘atinet¸u-Giurgiu and Neculai Stanciu W6. Ifa, b, c are the sides of a triangle, demonstrate the inequality

b+ca + _c+a^b +_a+b^c + _R^r ≤2.

St˘anescu Florin W7. If all triangle ABC holds

�sinA−�

sinA≥�

sin³A≥�

sinA+ 4�

cosA(� sinA).

St˘anescu Florin W8. Let f, g: [0,1]→(0,+∞), f(0) =g(0) = 0 two continuous functions such thatf it’s convex , and gconcave. Ifh: [0,1]→R is an increasing function, show that

�1 0

h(x)g(x)dx

�1 0

f(x)dx≤

�1 0

g(x)dx

�1 0

h(x)f(x)dx.

St˘anescu Florin W9. Let nbe a positive integer. Prove that

�n

k=1

F_2k

(F_2k+1−1)² <2,

whereF_n is then^thFibonacci number. That is, F₀= 0, F₁= 1 and F_n+2=F_n+1+F_n forn≥1.

Jos´e Luis D´ıaz-Barrero W10. Let a, b,andcbe positive real numbers. Prove that

�(6n+ 1)a−b n(b+c)

�₂ +

�(6n+ 1)b−c n(c+a)

�₂ +

�(6n+ 1)c−a n(a+b)

�₂

≥27 for any positive integern≥1.

Jos´e Luis D´ıaz-Barrero

W11. Let x > y > z > tbe four positive integers such that (x²−y²) + (xz−yt)−(z²−t²) = 0.

Prove thatxy+zt is a composite number.

Jos´e Luis D´ıaz-Barrero W12. Let n∈N and letOn = 1 +¹₃ +...+_2n−1¹ . Calculate

n→∞lim

1n

�1 +^2O_nⁿ�n

.

Ovidiu Furdui W13. Let (a_n)_n∈N be a sequence of real numbers such that

n→∞lim n(a_n−1) =l∈(−∞,∞) and letp≥1 be a natural number. Calculate

n→∞lim

�n k=1

�

a_n+ √p¹

kn

� .

Ovidiu Furdui W14. 1). Let f, g: [a, b]→Rbe two nonnegative continuous functions.

Assume that f attains its maximum at a unique point on [a, b] andg attains its maximum at the same point asf and possibly at other points. Prove that

n→∞lim

�b a

fⁿ⁺¹(x)g(x)dx

�b a

fⁿ(x)dx

=�f�_∞�g�_∞

2). Does the result hold under no assumption on f andg?

Ovidiu Furdui W15. Let a, b, c be positive real numbers. Prove that

�

cyc

� 2a²

a+b+ a³ b²+c²

�

≥ 9 2

a²+b²+c² a+b+c

Paolo Perfetti W16. Prove that sint�

cos³t+√ sint�

≥�_2t

π

�cos²t

when 0≤t≤ ^π₂.

Paolo Perfetti

W17. Let pa positive real number and let{an}_n≥1 be a sequence defined by a1 = 1, an+1= _1+a^anp

n.Find those real valuesq�= 0 such that the following series converges �^∞

n=1

��

�(pn)^−1p−a_n

��

�^q.

Paolo Perfetti W18. In 3-dimensional Euclidean space, let S be a sphere with centreO and radius R. Three pairwaise orthogonal rays originating in O intersectS atA, B, C. LetP be a point of S and leta, b, cdenote the areas of triangles OP A, OP B, OP C,respectively. Prove

2a(+b+c)�

a³+b³+c³�

≥R⁴(ab+bc+ca).

Michael Battaille W19. Let a, b, c, x, y, z be real numbers such that a, b, c >0 and

a+b+c=x+y+z= 1. Letu=xy−bz, v =az−cx, w=bx−ay and M =





2y

b+c z−c b−y c−z _c+a^2v x−a y−b a−x _a+b^2w



. Prove that det (M) = 0 if and only if u+v+w = 0.

Michael Battaille W20. Let Γbe a circle and let A∈Γ, B∈Γ, C /∈Γ be such thatCAand CB intersectΓagain at diametrically opposite points. Ifl is the radial axis ofΓand the circle with centre A, radius AC, show that d(C, l) = ^CA·CB_AB . (d(C, l) denotes the distance fromC to l).

Michael Battaille W21. Let ABC be a triangle and we noteAB=c, BC=a, CA=b and m_a, m_b, m_c the medians lengths corresponding to the vertexesA, B respectiveC.Prove the inequality �(^b2+c2)²

m³_a ≥ ³²₉^√3(a+b+c).

Ovidiu Pop W22. Let ABCD be a convex quadrilateral, O∈[AC], OM �BC,

M ∈AB, ON �AB, N ∈BC, OP �AD, P ∈CDandOQ�CD, Q∈DA.

Prove the inequalities min{Area[ABD], Area[BCD]}≤Area[M N P Q]≤

≤max{Area[ABD], Area[BCD]}.

Ovidiu Pop W23. Let n∈N, for k integer, 1≤k≤n,euclidean division nbyk gives n=qk+n_k,and denote pn the probability that n_k ≥ ^k₂. Calculate pn and find lim

n→∞pn.

Moubinol Omarjee W24. Let f ∈C³(Rⁿ, R) with f(0) =f^′(0) = 0. Prove that there exist h∈C³(R^m, S_n(R)), such thatf(x) =x^th(x)x, when S_n(R), is the set of symmetric matrix, and x^t is the transpose ofx.

Moubinol Omarjee W25. Find the nature of the series �

n≥1 e^iln(pn)

pn when (p_n)_n≥1 is the prime number increasing order, andi imaginary complex number.

Moubinol Omarjee W26. Let M be a point in the interior of triangle ABC andRa, R_b, Rc the radii of circumcircle ofM BC, M CA, M AB.Show that

1 Ra + _R¹

b +_R¹

c ≤ _{M A}¹ +_{M B}¹ +_{M C}¹

Nicu¸sor Minculete W27. Let aj >0, (j= 1,2, . . . , k) such that �

cyclic k−1�

j=1

aj =k, and n >1.

Prove that �

cyclic

n

�

a1+ 1

�_k

j=1a_j ≥k√ⁿ 2.

Angel Plaza´ W28. Let (xn)_n≥0 be the sequence defined recurrently by

x_n+2=x_n+1− ¹₂x_n with initial terms x₀= 2 andx₁ = 1. Find

�∞

n=1

xn

n+ 2. Angel Plaza´

W29. Let x, y, z be positive real numbers such thatx+y+z= 1. Prove that

�

x³+ 1 x²+y+z +

�

y³+ 1 y²+z+x+

�

z³+ 1

z²+x+y ≤3√ 2.

Angel Plaza´ W30. Let be x0=x1= 1 andxn+2= 5xn+1−xn−1 for all n≥0. Prove x_nx_mx_px_m+1x_m+2x_p+2≥

�

3

�

x²_n+1x_m+1+ ³

�

x²_m+1x_p+1+�³

x²_p+1x_n+1

�₃ for alln, m, p∈N.

Mih´aly Bencze W31. Ifa_k >1 (k= 1,2, ..., n) then � �

log_a1a₁a₂�_λ

≥n2^λ for all λ∈(−∞,0]∪[1,+∞).

Mih´aly Bencze W32. Ifx, y, z >0 and x+y+z= 1 then

8 (1−x)^x(1−y)^y(1−z)^z≤3 (2−x−y)^x+y(2−y−z)^y+z(2−z−x)^z+x. Mih´aly Bencze W33. 1). Let f :R→R be a bijective and continuous function such that f(a) =λaandf(b) =λbwhen a, b∈Randλ∈R^∗. Prove that exist x0∈(a, b) such thatf(x0) +λf⁻¹(λx0) = 2λx0.

2). Letf :R→Rbe a bijective and continuous function such that f(a) =λbandf(b) =λa when a, b∈Randλ∈R^∗. Prove that exist x₀∈(a, b) such thatf

�f(x0) λ

�

=λx₀.

Mih´aly Bencze W34. Prove �ⁿ

k=1

�2(k+1)(k+2)² ((k+2)!)³

�_k¹

≥ 3(n+2)(n+3)ⁿ⁽ⁿ⁺⁵⁾ .

Mih´aly Bencze W35. Prove that exist infinitely manyn∈N for whichn! is divisible by n⁵+n−1.

Mih´aly Bencze

W36. Let △(x, y, z) = 2 (xy+yz+zx)−�

x²+y²+z²�

and leta, b, cbe sidelengths of a triangle with areaF. Prove that△�

a³, b³, c³�

≤ ^64F√₃³. Arkady Alt W37. Let E be a inner Product Space with dot product −·−andF be proper nonzero subspace. LetP :E→E be orthogonal proiection E on F.

a). Prove that for any x, y∈E, holds inequality

|x·y−xP(y)−yP(x)|≤ �x�·�y�

b). Determine all cases when equality occours

Arkady Alt W38. Prove that 0<�₄x+2^x+1

x

�_x

−2^x<1 for all x∈� 0,_2e¹�

.

Ionel Tudor W39. Let n≥2 be a natural number anda_i>0, i= 1, n. If S=�_n

i=1a_i andxi=S−ai, then the following inequality holds:

�_n

i=1

√ai

(n−1)��

1≤i<j≤n(a_i+a_j) ≤

�_n

i=1

√xi

(n−1)��

1≤i<j≤n(x_i+x_j).

Ovidiu Bagdasar W40. Prove that ifx_i>0, i= 1, n,then the next inequality holds:

�n

i=1

S_α+β−x^α+β_i

S_α−x^α_i ≤n· S_α+β

S_α , (33)

provided thatαβ ≥0 and Sp =�_n

i=1x^p_i,for any real number p.

Ovidiu Bagdasar W41. Let n≥2 a natural number and the numbersai>1, i= 1, n. Prove that

�n

i=1

log_aiaⁿ⁻¹_i+1

S−a_i ≥ n²

�_n

i=1a_i.

We consider thatan+1=a1, andS= �ⁿ

i=1

ai.

Ovidiu Bagdasar W42. Let ABC be an acute triangle. The angle bisectors fromA, B, C meet the opposite sides inA1, B1, C1, respectively. LetRandr be the circumradius and the inradius of the triangleABC,respectively. Let RA, R_B,andR_C the circumradii of the triangles AC₁B₁, BA₁C₁, andCB₁A₁, respectively. Prove that

R_A+R_B +R_C≥R+r.

P´al P´eter D´alyay W43. Let f be a continuous real function defined on the set of the

nonnegative real numbers for which the following integrals are convergent:

S=�_∞

0 f²(x)dx, T =�_∞

0 xf²(x)dx, U =�_∞

0 x²f²(x)dx. Prove that

�� _∞

0 |f(x)|dx

�₂

≤2

�√

SU+T

� .

P´al P´eter D´alyay W44. Ifζ is the Riemann zeta-function, ands is a real number greater than 3/2, then:

ζ²(s)≤π√ 2





�∞

i,j=1

1 i^s−1j^s−1(i+j)

�∞

k,l=1

1 k^s−1l^s−1(k+l)³





1/2

≤

≤ π 2√

2ζ(s−1/2)ζ(s+ 1/2).

P´al P´eter D´alyay W45. Let x∈P oisson(2) be a random variable

i). Find the setM of all the valuesn∈N^∗ so that P��

ω/|x(ω)−2|≥ ²_n��

≤ ¹²⁸_n2

ii). From the set M we extract 2 numbers one, after the other. Find the probability that the second extracted number could be the greatest, in the following two situation: with return in the set, and without return in M.

Laurent¸iu Modan

W46. i). For any n >3 natural numbers, prove that 2n! + (2n)!>3·2ⁿ⁺² ii). Study the convergence of the series �

n>3 1 2n!+(2n)!

Laurent¸iu Modan W47. Let a, b, c, d∈(0,1) and f : (0,1)→Ra convex and decreasing function. Prove that�

f� 1−a³�

≥� f�

1−a²b� .

Marius Dr˘agan W48. Let I be an interval andf :I→Ra convex function and

x₁, x₂, ..., x_n∈I.Prove that �ⁿ

k=1

f(x_k)−nf

�

1 n

�n k=1

x_k

�

≥

1≤i<...<imaxk≤n

�

f(x_i1) +...+f(x_ik)−kf�_x

i1+...+x_ik k

��

Marius Dr˘agan and Mih´aly Bencze W49. Find all the functionsf :R→[−1,1] such that

f(x+y) (1−f(x)f(y)) =f(x) +f(y) for each x, y∈R.

Marius Dr˘agan and Sorin R˘adulescu W50. Find all the function f :R→Rwhich are continuous in a real point x0 such that f(x+y) =f(x)�

1 +f²(y) +f(y)�

1 +f²(x) for each x, y∈R.

Sorin R˘adulescu